Suppose that, a bank account that earns 10% interest compounded continuously has an initial balance of zero, i.e. and money is deposited into the account at a constant rate of per year.
To set up a differential equation for the net worth, use the fact that
Rate net worth increasing = Rate revenue is earned + Rate deposits are made
Revenue is earned at a rate of 10% i.e. and deposits are made at a rate of 1000 dollars per year.
So, for in years,
To find the balance as a function of time, it is need to solve the differential equation .
To find the solution of the differential equation, use method of separation of variables by keeping all theterms on one side of the equation and remaining are on the other.
Then integrate each side separately:
for some constant
Solving for leads to
Exponential of a quantity is always positive, so is positive.
Where , so is any nonzero constant.
Therefore, the general solution is
To find the value of, use the initial condition
That is, substitute and into the general solution,
Hence, the required solution is